Rigging an Umbrella and Weight Paint
I figured it out. The initial mesh cannot be subdivided more than the number of bone ribs. Each vertex should be mapped to a bone. Watching this tutorial on rigging an umbrella in 3ds Max gave me the answer.

I had basically overcomplicated the problem by subdivisions assigned with weights in between the two ribs.

In other words, I have 6 ribs for the umbrella, with 4 bones along the umbrella mesh. Each bone gets mapped to a vertex.